Using a sample of $(225.3\pm 2.8)\times 10^{6}$ $J/\psi$ decays collected with the BESIII detector at BEPCII, searches for invisible decays of $\eta$ and $\eta^\prime$ in $J/\psi\to\phi\eta$ and $\phi\eta^\prime$ are performed. Decays of $\phi \to K^{+}K^{-}$ are used to tag the $\eta$ and $\eta^\prime$ decays. No signals above background are found for the invisible decays, and upper limits at the 90% confidence level are determined to be $2.58\times10^{-4}$ for the ratio $\frac{\mathcal{B}(\eta\to\rm{invisible})}{\mathcal{B}(\eta\to\gamma\gamma)}$ and $2.39\times10^{-2}$ for $\frac{\mathcal{B}(\eta^\prime\to\rm{invisible})}{\mathcal{B}(\eta^\prime \to\gamma\gamma)}$.

Comment: 8 pages, 5 figures, and 2 tables, add more reference, changes made in introduction and conclusion Accepted by Phys. Rev. D; this the version for publication at PRD